Date of Award


Degree Type


Degree Name

Doctor of Philosophy


Let M be a reductive linear algebraic monoid with unit group G and let the derived group of G be simply connected. The purpose of this thesis is to study the centralizer in M of a semisimple element of G. We call this set {dollar}M\sb0.{dollar};We use a combination of the theories of algebraic geometry, linear algebraic groups and linear algebraic monoids in our study. One of our main tools is Renner's analogue of the classical Bruhat decomposition for reductive algebraic monoids. Our principal result establishes an analogue of the Bruhat decomposition for {dollar}M\sb0.{dollar} This is a more general result than Renner's decomposition for the centralizer of a torus on a reductive algebraic monoid.;Early research by M. S. Putcha and L. E. Renner presents the basic notation and general theory of algebraic monoids and is mainly descriptive. Later interest centres around the theory of reductive algebraic monoids which, by definition, are always irreducible. In this thesis we investigate the irreducibility of {dollar}M\sb0.{dollar} After proving propositions about the structural properties of {dollar}M\sb0,{dollar} we give a characterization of the irreducibility of {dollar}M\sb0.{dollar};Finally, we give examples of algebraic monoids that have only irreducible centralizers and one in which the centralizer {dollar}M\sb0{dollar} is reducible.



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